# What is the probability the student has a score less than 25.4, and what is the value of `b`? The scores of a test given to students are normally distributed with a mean of 21. 80% of the students have scores less than 23.7. The standard deviation of the scores is 3.214. A student is chosen at random. This student has the same probability of having a score less than 25.4 as having a score greater than `b`.

To answer the first part of the question, you need to find a table containing values for normal distribution. Usually, this will be a table of z-scores, which are calculated in the following way:

`z = (x-mu)/sigma`

Here, `x` is the sample value, `mu` is the global mean, and `sigma`...

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To answer the first part of the question, you need to find a table containing values for normal distribution. Usually, this will be a table of z-scores, which are calculated in the following way:

`z = (x-mu)/sigma`

Here, `x` is the sample value, `mu` is the global mean, and `sigma` is the global standard deviation. Let's calculate the z-score for a score of 25.4:

`z = (25.4-21)/3.214 = 1.37`

So, 25.4 is 1.37 standard deviations above the mean. Using the z-table found in the link below, we can now say that 0.913 of the population will be below this score. Therefore, the probability that a student has a score less than 25.4 is 0.913.

Using symmetry of the normal distribution, we can say that if the probability of a student having a score greater than b has the same probability as a student having a score less than 25.4, then the z-score will be reflected across zero. In other words, the z-score for b will be the negative of the z-score for 25.4. this gives us the following z-score:

`z_b = -1.37`

To find b with the z-score, we simply substitute all of our known variables into the z-score formula and solve for the sample value:

`z_b = (b-mu)/sigma`

`-1.37 = (b-21)/3.214`

Multiply both sides by 3.214:

`-4.40318 = b-21`